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I have set up an external content source, crawled the data and set up a search scope which is successfully returning results from the external source.
How can that external content source (website) make use of the same search functionality to display a search box and results on its own pages?
You can write either javascript or other code to call the SharePoint Search Query Web Service and post a query and return results. If you move to SharePoint 2013 you can use REST for the request/response.
Once you get the results back it is up to you to build the result user interface. There are tons of examples if you search for them.
